Modal Analysis

Download Modal Analysis 1.0.0 for Android
Download Modal Analysis 1.0.0 for Windows
Download Modal Analysis 1.0.0 for Mac, Linux (.jre is required)

Modal Analysis is an Android and Desktop application that provides the circular frequencies (ω), modal periods (T) and eigenvectors (φ) of elastic dynamic systems with multiple degrees of freedom.

It can be launched in every Android device and any OS (Windows, Mac, Linux) and the only user provided data, necessary for the application to run, are:

1. The system's degrees of freedom (1 ~ 6)
2. The Mass matrix
3. The Stiffness matrix

The application does the rest for you!

Methodology

The application uses the Generalized Jacobi method to determine the eigenvalues and eigenvectors of the defined problem. This methodology is presented in detail in [1] and can be summarized as follows:

Step 1

Firstly, the eigenvector is assumed to be equal to [I] and the eigenvalues, with the portion of the diagonal elements of the stiffness and mass matrix:

Step 2

The off-diagonal elements of [K] and [M] are checked, in order to determine whether they satisfy the below inequalities:

If an element satisfies one of the above, the program proceeds to Step 3, otherwise it checks next element.

Step 3

The rotational parameters a and c are calculated as follows

In case b=0 then:

Step 4

The orthogonal matrix [Q] is declared and a generalized rotation is performed, so for the off-diagonal element (i,j) to become zero and as a result, new stiffness ( [Knew] ) and mass ( [Mnew] ) matrices to be formed.

Step 5

The eigenvector matrix is updated, using the [Q] matrix as below:

Step 6

When all the non zero off-diagonal elements are eliminated, the eigenvalues are also updated:

Step 7

The eigenvalues are checked for convergence

if this inequality is not satisfied, the procedure starts again from Step 2 with updated eigenvalues and stiffness-mass matrices.

Step 8

The new diagonal matrices are checked for convergence

if one these inequalities is satisfied, the procedure starts again from Step 2

Step 9

Once convergence is achieved, the eigenvectors are scaled by the diagonal mass matrix and the procedure is concluded by retrieving and sorting the final eigenvectors and eigenvalues of the problem.

References

[1]  Doyle JF. STATIC AND DYNAMIC ANALYSIS OF STRUCTURES with An Emphasis on Mechanics and Computer Matrix Methods. Dordrecht, Netherlands: Springer; 1991. doi:10.1007/978-94-011-3420-0.